Spectral Approximation: Identifying classes of operator for which efficient spectrally inclusive algorithms can be devised

Operators on infinite-dimensional spaces arise in numerous physical applications, including quantum chemistry, structural mechanics, design of electrical circuits and photonic waveguides. The physical behaviours of all these systems depend on the spectra of the operators concerned. However the calculation of the spectra involves some intermediate reduction, often to a large matrix, and this presents serious challenges for spectral approximation. Traditionally the main problem has been spectral pollution, i.e. the approximations generate sequences of eigenvalues which converge to points not in the spectrum of the original operator. However, there are also dramatic examples in which spectral inclusion may fail, a problem which currently can only be addressed by expensive pseudo-spectral techniques. Meanwhile, there are at least two good schemes for filtering or avoiding spectral pollution.

This project will deal with problems of spectral approximation for operators and operator pencils and try to identify classes of operator for which efficient spectrally inclusive algorithms can be devised. Typical examples will include elliptic partial differential operators, block operator matrices, discrete operators in $\ell^2$ and multiplication operators; if time permits, we shall also consider operators on singular manifolds.